Fractal FoodSelf-Similarity on the Supermarket Shelf

Fractal forms—complex shapes which look more or less the same at
a wide variety of scale factors, are everywhere in nature. From the
fluctuations in the cosmic
microwave background radiation to the coastlines of continents,
courses of rivers, clouds in the sky, branches of plants and veins in
their leaves, blood vessels in the lung, and the shape of seashells
and snowflakes, these fractal or self-similar patterns
abound. The self-similarity of most of these patterns is defined only
in a statistical sense: while the general “roughness” is
about the same at different scales, you can't extract a segment, blow
it up, and find a larger scale segment which it matches precisely.

However, some of the most pleasing patterns in geometric art exhibit
exact or almost exact self-similarity. These are patterns which
are composed of smaller copies of themselves
ad infinitum, or at least until some
limit where the similarity breaks down
due to the granularity of the underlying material.

The Unclassifiable Romanesco

Nearly exact self-similar fractal forms occur do in nature, but I'd
never seen such a beautiful and perfect example until, some time
after moving to Switzerland, I came across a chou
Romanesco like the one above in a grocery store. This is so
visually stunning an object that on first encounter it's hard
to imagine you're looking at a garden vegetable rather than an alien artefact
created with
molecular
nanotechnology. But of course, then you
realise that vegetables are created with molecular
nanotechnology, albeit the product of earthly evolution,
not extraterrestrial engineering.

Perhaps it's in part due to how alien this vegetable appears that
there's so little consensus as to what it should be called.
Romanesco (as I'll refer to it henceforth) is a member of the species
Brassica oleracea L., which includes cabbage, broccoli,
Brussels sprouts, cauliflower, collard greens, kohlrabi, and numerous
other “cultivars” (cultivated variations). Plant species are broader
and more diverse than those of animals. All of these plants,
notwithstanding their gross morphological differences, can and have
been crossed, resulting in such innovations as broccolini (a cross of
Chinese kale and broccoli) and broccoflower (a cross of broccoli and
cauliflower which superficially resembles Romanesco but lacks its
near perfect self-similar fractal form). I wonder what you'd get if
you crossed red cabbage with Romanesco? Hmmm….

The French name, chou Romanesco literally
translates to “Romanesco cabbage”, placing it in the cabbage family
even though it doesn't much resemble any cabbage you've ever seen.
In German, it's Pyramidenblumenkohl: “pyramid
cauliflower”; in Italy, where it was first described in the sixteenth
century, it's called broccolo romanesco:
“Romanesco broccoli”, but sometimes cavolo
romanesco: “Romanesco cabbage”. Finally, in English it's usually
called “Romanesco broccoli”, but you'll also see it referred to as
“Romanesco cauliflower”. Even professional plant taxonomists can't
decide precisely where it belongs; some place it within the
Italica group with broccoli,
while others argue
it belongs in the Botrytis group with cauliflower.
Broccoli, cabbage, cauliflower—beats me—let's just consider
it sui generis and call it “Romanesco”.

A Computational Universe?

These natural fractal patterns, of great apparent complexity, can be
simulated by simple computer programs such as our
Terranova, its
companion
Terranova Screen Saver,
and
Cellular Automata Laboratory, producing
results which mimic those in nature. This tempts one to speculate
that nature generates these patterns through a process akin to
computation.

It seems like the universe just wants to compute. Of
course, there's a tendency for thinkers in every age to model the
universe in terms of the predominant technology of the day. To the
Pythagoreans, all was number and geometry. In Newton's time, the
universe seemed an intricate clockwork mechanism. Later, in the age
of steam, thermodynamics and heat death dominated models of the
universe. Today, surrounded by computers evolving more rapidly than
anything in natural history, what could be more natural than
regarding the universe as a great automaton performing some kind of
cosmic computation?

And yet, there may be some truth in that viewpoint, and insights to be
had by pursuing it, just as earlier worldviews provided
frameworks for further discoveries.
Stephen Wolfram's
A New Kind of Science
and Rudy Rucker's forthcoming
The Lifebox, the Seashell, and the Soul
(excerpt)
argue that many of the processes we see in nature are indeed
computations.

Wolfram finds that, essentially regardless of details, the results of
iterated computations fall into four general (although not entirely
exclusive) classes. Class 1 computations produce uniform results
from almost any input. Class 2 computations produce output which
depends upon the input, but the results either stay the same forever
or repeat with a short cycle time. Class 3 computations produce
output which appears random (and often passes stringent tests of
randomness), while Class 4 computations balance on the edge of order
(Class 2) and chaos (Class 3), manifesting localised structures which
move and interact with one another in complicated ways. Starting a
one-dimensional
cellular automaton with random
input and various rules demonstrates the behaviour of the four
classes of computation.

Class 1: Rule 250

Class 2: Rule 132

Class 3: Rule 122

Class 4: Rule 110

Many Class 3 computations produce self-similar or
fractal output, which Wolfram refers to as “nested”.
The image below is produced by a computer program
eight bits in length—the number 126—interpreted
to define the new state of a cell based on its current
state and those of its two immediate neighbours. The program
is started on the the top line of the image, which consists of
a single black cell in the middle of the line. Subsequent lines
show the evolution as the program is applied over and over,
each line serving as input to the line below it.

The structure produced by this rule was named the “Sierpiński
Gasket” by Benoit Mandelbrot; the same pattern appears in Pascal's
triangle of binomial coefficients. Note the intricate nesting of the
white triangles; in this small image, eight levels of nesting are
present (counting the partial triangle at the bottom). Extrapolated
to infinity, an infinite number of nesting levels will be
present. Heck of a lot to get from a computer program you can write
down as a three digit decimal number, don't you think? Even though
this is a simple two dimensional pattern produced by a one
dimensional computation, the similarity with the three dimensional
hierarchical structure of the Romanesco is compelling.

Stalking the Vegetable with a Camera

These photos were all taken with a Nikon D70 digital camera. The photo above
of the entire Romanesco was taken with a NIKKOR AF 28–80 mm zoom lens at 44 mm
(note that due to the size of the image sensor in the D70, focal lengths should
be multiplied by a factor of 1.5 for the 35 mm film camera equivalent).
Lighting was a mix of natural and overhead fluorescent light, arranged (not
entirely successfully) to minimise shadows. Exposure was ½ second at f/29.

The close-up photos were all taken with a vintage Micro-NIKKOR
55 mm f/2.8 manual focus macro lens, stopped down to the minimum
aperture of f/32 to maximise depth of field; at f/32, the
hyperfocal
distance is such that everything from the closest point at which the
lens can focus to more than four metres is effectively in focus.
The first photo below was taken with the macro lens mounted directly
on the camera body with a 4 second exposure time, while the balance
of the photos used a Nikon PK-12 14 mm extension tube between
the camera and lens to permit focusing even closer than the minimum
25 cm object to image plane distance of the macro lens alone.
With the macro lens and extension tube, it's possible to focus as
closely as 21 cm object to image plane, at which point the front
of the lens barrel is only about 7 cm from the object. The extreme
close-ups taken with the extension tube required an exposure of 13
seconds at f/32. Fortunately, as photographic subjects, vegetables,
even fractal ones, aren't nearly as fractious as supermodels or
cows,
so such long exposure times pose no difficulty as long as the camera
is mounted on a sturdy tripod. All of the close-ups were lit entirely
by overhead fluorescent lights.

Images were postprocessed with The
Gimp on Linux. Postprocessing amounted to cropping, modest sharpening, and
adjusting the colour balance to approximate the actual colour of the vegetable
as perceived by the human eye under natural light. When you shoot a photo
like the close-ups where the entire field of view is a uniform hue like
green, the automatic white balance in the camera will shift the white point
to try to adjust the picture toward the white. It's best to disable
the automatic white balance entirely, but if you forget to (as I did when taking
these shots), it's easy enough to correct after the fact. Yes, a Romanesco
is actually the radioactive green colour shown in these pictures. The leaves
are a darker blue green typical of broccoli or cauliflower. When cooked,
the colour lightens to a less saturated greenish white.

Every self-similar pattern in nature breaks down at some scale—at
the level of molecules and atoms if not before. The last photo
shows the tiny structures near the top level spiral. As the
spirals get smaller and smaller approaching the
vertex, the spirals that make them
up have less and less lower level detail, with the
tiniest being little more than bumpy spheroids.

Fixing Fractal Food

Romanesco is excellent raw, enhancing both the appearance and taste
of an assiette de crudités. It's crunchier than
cauliflower and not as bland. It has a nutty taste (and looks kind
of nutty too until you get used to it!) and doesn't have the chalky
edge which some people dislike in broccoli. Any dip that's good with
cauliflower and broccoli will go fine with Romanesco, but be sure to
try it by itself—you may decide to forgo the dip. It would be
absolutely ideal to serve raw Romanesco on a platter with an
image of the Mandelbrot set!

Romanesco can be cooked using any method that's suitable for
broccoli or cauliflower, and may be substituted in any recipe
which calls for them. My personal favourite way to prepare it
it to break off the “level 1” spirals (it's easier to do this
with the ones at the base if you first cut them loose from the
central stem by running a short knife around it from the bottom),
then steam them for between 15–25 minutes depending on how
crunchy you like your vegetables. Steaming preserves far more
of the vitamins in vegetables than boiling, and doesn't tend
to reduce their colour to a uniform grey.

If you're counting
calories, figure 34 (kilo)calories (134 kilojoules) per 100
grams of Romanesco, almost precisely the same as broccoli and
cauliflower; note that there are as many calories in a single pat of
butter! Romanesco is rich in Vitamin C, folic acid, potassium, and
fibre. A typical Romanesco weighs between 300 and 600 grams.

Cellular Automata Image Maker

You can
download the program used to
create the cellular automata images which appear in this page.
This Perl program produces portable bitmap images which
you can postprocess with the Netpbm image processing
toolkit. This program is in the public domain; you can use it in
any way you wish without restrictions, but it is utterly
unsupported—you are entirely on your own.